DNMW04 - New trends and challenges in the mathematics of optimal design

The deshomogenization method consists in reconstructing a minimization sequence of genuine shapes converging toward the optimal composite. We introduced this method a few years ago. Since, it has gain some interest—see the works of JP. Groen and O. Sigmund—thanks to the rise of additive manufacturing. Bascillay, it can be considered as a post-treatment of the classical homogenization method. The output of the (periodic) homogenization method is : – An orientation field of the periodic cells – Geometric parameters describing the local micro-structure. From this output, the deshomogenization method allows to construct a sequence of genuine shapes, converging toward the optimal, (almost) suitable for 3D printers.

The sequence of shapes is defined via a so called “grid map”, which aim is to ensure the correct alignment of the cells with respect to the orientation. field. It also enforce the connectivity of the structure between neighboring cells. If the orientation field is regular and the optimization domain $D$ is simply connect, the grid map can be defined as local diffeomorphism from $D$ into $R^n$ (with n=2 or 3). If those requirements are not met, the definition of the grid map is much more intricate.

Moreover, a minimal kind of regularity is needed to be able to ensure the convergence of the sequence of shapes toward the optimal composite : it is necessary to regularize the orientation field but still allow for the presence of singularities. This is done by a penalization of the cost function based on the Ginzburg-Landau theory.

In this talk, we will present 1/ A general definition of the grid map based on the introdcution of an abstract manifold. 2/ A regularization of the orientation field based on G-L theory. 3/ Numerical applications in 2D and 3D.

This talk is based on a joint work by G. Allaire, P. Geoffroy and K. Trabelsi.